Introduction to Statistics for Astronomers and Physicists
Section 0: Introduction, Lecture Outline, Course Format
Angus H Wright
2021-03-22
Setup & Course Outline
This course will be taught in 4 parts, each spanning from 2-4 weeks
Section 1: Data Description, Analysis, and Modelling (Weeks 1-2)
Topics include:
- Types of data
- Point & interval estimation
- Correlation & covariance
- Fundamentals of data exploration/mining
- Introduction to data visualisation
##Section 2: Probability & Decision Making (Weeks 3-5)
Topics include:
- Decision theory
- fundamentals of probability
- statistical distributions and their origins
##Section 3: Bayesian Statistics (Weeks 6-8
Topics include:
- Frequentist & Bayesian statistics
- Bayes theory
- prior specification
- hypothesis testing,
##Section 4: Parameter Simulation, Optimisation, and Analysis (Weeks 9-12)
Topics include:
Course Philosophy
This course is designed to be a practical introduction to statistics for astronomers and physicists who are starting their research careers and have had little (or perhaps no) previous education in statistics and statistical data analysis. The course (and these lecture notes) are not designed to be a statistics reference text. Rather the material presented here is designed to guide students on a suitable path towards robust data analysis and research.
The course will present many aspects of data analysis that are widely relevant to modern astronomy and physics. We will borrow heavily from standard statistical problems and thought experiments in an effort to convey important points, and elucidate common statistical and logical fallacies. Problems will almost always be explored using a mixture of tools simultaneously: plain English, math, computer code, graphs, and more.
Rmarkdown
Slides and lecture notes for this course are prepared in Rmarkdown, and provided to you after the lectures.
The utility of
Rmarkdown is that it allows running execution of code chunks alongside markdown-style text, in a wide array of languages, including R, Python, bash, Rcpp, javascript, and more. This allows us to present examples in multiple languages easily within one document. For example, if I want to plot a function, I can do so in:
#in R
theta=seq(0,2,by=0.01)
sinu=1+sin(2*pi*theta)
magplot(theta,sinu,type='l')
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#or in python
import numpy as np
import matplotlib.pyplot as plt
theta=np.arange(0.,2.,0.01)
sinu=1+np.sin(2*np.pi*theta)
plt.plot(theta,sinu)
plt.show()
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Information generated and stored within blocks is persistent, and code-blocks with different engines can also cross-communicate. This means that, for example, we can:
#Create some data in R
#E.g. random draws from f~N(0,1)
x=rnorm(1e3)
y=rnorm(1e3)
#Default plot in R
plot(x,y)
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#and access it directly in python
plt.scatter(r.x,r.y)
plt.show()
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Modern Science and the Requirement of Programming
Modern physics and astronomy requires an understanding of programming. From theoreticians writing models to experimentalists writing analysis pipelines, most physicists and astronomers will use read, write, or use a computer program every day.
The ubiquitousness of programming in the modern physical sciences is intimately linked to the important role that statistics plays in these fields. Modern science is increasingly reliant on large and/or complex datasets, which fundamentally must be analysed using modern statistical methods. A classic example is model optimisation (which we will cover in detail in this course): let us take a relatively simple dataset that we know follows a generic beta distribution, and attempt to model this dataset using a function containing 2 parameters:
\[ Y = Beta(X, \alpha,\beta) + {\rm noise} \\
0\leq X \leq 1 \\
\alpha, \beta \in [0,\infty)
\]
One might be inclined to attempt to fit a model to these data by-hand, using trial and error:
Using this approach we can get a reasonable fit with \(\alpha=2.8,\beta=6.7\):
But is this solution near the truth? How close is good enough? And what are the uncertainties on the parameters? These are all important in modern science, and they are precisely the sort of questions/problems that computers/programs are designed to tackle. With just one command, we can use R/python to reach a more accurate solution, in a fraction of the time.
#Estimate parameters using Nonlinear Least Squares (nls) in R
fit_R=nls(y~dbeta(x,alpha,beta), #The function to fit
data=list(x=x,y=y), #The data
start=list(alpha=2,beta=5), #The initial guess
algorithm='port', #The algorithm
lower=c(0,0)) #The lower bounds
best_R=summary(fit_R)$parameters
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#Estimate parameters using scipy.optimize.curve_fit in python
import scipy.optimize
import scipy.stats
best_py, cov_py = scipy.optimize.curve_fit(
scipy.stats.beta.pdf, #The function to fit
r.x, r.y, #The data
p0=[2,5], #The initial guess
bounds=(0,np.inf),#The lower and upper bounds
method='trf') #The fitting algorithm
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Obviously these fits are superior to those which we can reach by-hand in terms of accuracy, effort, and runtime. But the most important benefit is in terms of uncertainty estimation. Statistical computing is important for a wide range of reasons, but arguably the first and most important reason is for the computation of measures of uncertainty.
#Model statistics in R
summary(fit_R)
##
## Formula: y ~ dbeta(x, alpha, beta)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## alpha 3.21166 0.01453 221.0 <2e-16 ***
## beta 7.70586 0.03728 206.7 <2e-16 ***
## ---
## Signif. codes:
## 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.09854 on 998 degrees of freedom
##
## Algorithm "port", convergence message: both X-convergence and relative convergence (5)
|
#Model parameters and covariance in python
print(best_py,cov_py)
## [3.21166224 7.70585894] [[0.00021118 0.00050005]
## [0.00050005 0.00138966]]
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Do I need to know R or Python?
Despite what the internet will tell you, today there is very little separating the two languages in terms of functionality. Both languages can be run as a subprocess of the other, both have well developed tools for data analysis and machine learning, and both have a wealth of tutorials and guides to help new users enter the game. Overall, there is one major consideration that will (and should) drive your choice of which language to learn first: what languages do your colleagues use? Any perceived benefit or detriment of the languages will invariably be overwhelmed by whether or not you are able to share and discuss code together with your colleagues. So, this is mostly a case where joining the herd is probably the sensible choice.
##What’s the difference?
R was originally developed as a statistics and data analysis language, and so many data analysis tools are available natively within base R. Additional tools are available through the Comprehensive R Archive Network (CRAN), which contains over 10k packages that are all required to be fully documented. This means that if you want to perform a particular flavour of statistical analysis, there is a good chance that well developed code already exists within R to do it (or at least to get you started).
Python’s strength lies within the its use as a general programming language beyond data analysis. In comparison, python is Personally I code primarily in R, however a significant R and Python are
In practice, the vast majority of examples in this course will be programmed in R. However, as you can see from the above, if you can understand one, then you can probably understand both. In this lecture we will go through some examples of R and python code, so that you have an introduction to the important parts (and can follow along without much trouble).
Lecture 0: Introduction to Statistics
for (i in 1:10) plot(runif(1000), ylim = c(0, 1)) # for example
x = 5 # radius of a circle
For a circle with the radius 5, its area is 78.5398163.
Why do we need statistics?
## Sepal.Length Sepal.Width Petal.Length Petal.Width
## 1 5.1 3.5 1.4 0.2
## 2 4.9 3.0 1.4 0.2
## 3 4.7 3.2 1.3 0.2
## 4 4.6 3.1 1.5 0.2
## 5 5.0 3.6 1.4 0.2
## 6 5.4 3.9 1.7 0.4
## 7 4.6 3.4 1.4 0.3
## 8 5.0 3.4 1.5 0.2
## 9 4.4 2.9 1.4 0.2
## 10 4.9 3.1 1.5 0.1
## 11 5.4 3.7 1.5 0.2
## 12 4.8 3.4 1.6 0.2
## 13 4.8 3.0 1.4 0.1
## 14 4.3 3.0 1.1 0.1
## 15 5.8 4.0 1.2 0.2
## 16 5.7 4.4 1.5 0.4
## 17 5.4 3.9 1.3 0.4
## 18 5.1 3.5 1.4 0.3
## 19 5.7 3.8 1.7 0.3
## 20 5.1 3.8 1.5 0.3
## 21 5.4 3.4 1.7 0.2
## 22 5.1 3.7 1.5 0.4
## 23 4.6 3.6 1.0 0.2
## 24 5.1 3.3 1.7 0.5
## 25 4.8 3.4 1.9 0.2
## 26 5.0 3.0 1.6 0.2
## 27 5.0 3.4 1.6 0.4
## 28 5.2 3.5 1.5 0.2
## 29 5.2 3.4 1.4 0.2
## 30 4.7 3.2 1.6 0.2
## 31 4.8 3.1 1.6 0.2
## 32 5.4 3.4 1.5 0.4
## 33 5.2 4.1 1.5 0.1
## 34 5.5 4.2 1.4 0.2
## 35 4.9 3.1 1.5 0.2
## 36 5.0 3.2 1.2 0.2
## 37 5.5 3.5 1.3 0.2
## 38 4.9 3.6 1.4 0.1
## 39 4.4 3.0 1.3 0.2
## 40 5.1 3.4 1.5 0.2
## 41 5.0 3.5 1.3 0.3
## 42 4.5 2.3 1.3 0.3
## 43 4.4 3.2 1.3 0.2
## 44 5.0 3.5 1.6 0.6
## 45 5.1 3.8 1.9 0.4
## 46 4.8 3.0 1.4 0.3
## 47 5.1 3.8 1.6 0.2
## 48 4.6 3.2 1.4 0.2
## 49 5.3 3.7 1.5 0.2
## 50 5.0 3.3 1.4 0.2
## 51 7.0 3.2 4.7 1.4
## 52 6.4 3.2 4.5 1.5
## 53 6.9 3.1 4.9 1.5
## 54 5.5 2.3 4.0 1.3
## 55 6.5 2.8 4.6 1.5
## 56 5.7 2.8 4.5 1.3
## 57 6.3 3.3 4.7 1.6
## 58 4.9 2.4 3.3 1.0
## 59 6.6 2.9 4.6 1.3
## 60 5.2 2.7 3.9 1.4
## 61 5.0 2.0 3.5 1.0
## 62 5.9 3.0 4.2 1.5
## 63 6.0 2.2 4.0 1.0
## 64 6.1 2.9 4.7 1.4
## 65 5.6 2.9 3.6 1.3
## 66 6.7 3.1 4.4 1.4
## 67 5.6 3.0 4.5 1.5
## 68 5.8 2.7 4.1 1.0
## 69 6.2 2.2 4.5 1.5
## 70 5.6 2.5 3.9 1.1
## 71 5.9 3.2 4.8 1.8
## 72 6.1 2.8 4.0 1.3
## 73 6.3 2.5 4.9 1.5
## 74 6.1 2.8 4.7 1.2
## 75 6.4 2.9 4.3 1.3
## 76 6.6 3.0 4.4 1.4
## 77 6.8 2.8 4.8 1.4
## 78 6.7 3.0 5.0 1.7
## 79 6.0 2.9 4.5 1.5
## 80 5.7 2.6 3.5 1.0
## 81 5.5 2.4 3.8 1.1
## 82 5.5 2.4 3.7 1.0
## 83 5.8 2.7 3.9 1.2
## 84 6.0 2.7 5.1 1.6
## 85 5.4 3.0 4.5 1.5
## 86 6.0 3.4 4.5 1.6
## 87 6.7 3.1 4.7 1.5
## 88 6.3 2.3 4.4 1.3
## 89 5.6 3.0 4.1 1.3
## 90 5.5 2.5 4.0 1.3
## 91 5.5 2.6 4.4 1.2
## 92 6.1 3.0 4.6 1.4
## 93 5.8 2.6 4.0 1.2
## 94 5.0 2.3 3.3 1.0
## 95 5.6 2.7 4.2 1.3
## 96 5.7 3.0 4.2 1.2
## 97 5.7 2.9 4.2 1.3
## 98 6.2 2.9 4.3 1.3
## 99 5.1 2.5 3.0 1.1
## 100 5.7 2.8 4.1 1.3
## 101 6.3 3.3 6.0 2.5
## 102 5.8 2.7 5.1 1.9
## 103 7.1 3.0 5.9 2.1
## 104 6.3 2.9 5.6 1.8
## 105 6.5 3.0 5.8 2.2
## 106 7.6 3.0 6.6 2.1
## 107 4.9 2.5 4.5 1.7
## 108 7.3 2.9 6.3 1.8
## 109 6.7 2.5 5.8 1.8
## 110 7.2 3.6 6.1 2.5
## 111 6.5 3.2 5.1 2.0
## 112 6.4 2.7 5.3 1.9
## 113 6.8 3.0 5.5 2.1
## 114 5.7 2.5 5.0 2.0
## 115 5.8 2.8 5.1 2.4
## 116 6.4 3.2 5.3 2.3
## 117 6.5 3.0 5.5 1.8
## 118 7.7 3.8 6.7 2.2
## 119 7.7 2.6 6.9 2.3
## 120 6.0 2.2 5.0 1.5
## 121 6.9 3.2 5.7 2.3
## 122 5.6 2.8 4.9 2.0
## 123 7.7 2.8 6.7 2.0
## 124 6.3 2.7 4.9 1.8
## 125 6.7 3.3 5.7 2.1
## 126 7.2 3.2 6.0 1.8
## 127 6.2 2.8 4.8 1.8
## 128 6.1 3.0 4.9 1.8
## 129 6.4 2.8 5.6 2.1
## 130 7.2 3.0 5.8 1.6
## 131 7.4 2.8 6.1 1.9
## 132 7.9 3.8 6.4 2.0
## 133 6.4 2.8 5.6 2.2
## 134 6.3 2.8 5.1 1.5
## 135 6.1 2.6 5.6 1.4
## 136 7.7 3.0 6.1 2.3
## 137 6.3 3.4 5.6 2.4
## 138 6.4 3.1 5.5 1.8
## 139 6.0 3.0 4.8 1.8
## 140 6.9 3.1 5.4 2.1
## 141 6.7 3.1 5.6 2.4
## 142 6.9 3.1 5.1 2.3
## 143 5.8 2.7 5.1 1.9
## 144 6.8 3.2 5.9 2.3
## 145 6.7 3.3 5.7 2.5
## 146 6.7 3.0 5.2 2.3
## 147 6.3 2.5 5.0 1.9
## 148 6.5 3.0 5.2 2.0
## 149 6.2 3.4 5.4 2.3
## 150 5.9 3.0 5.1 1.8
## Species
## 1 setosa
## 2 setosa
## 3 setosa
## 4 setosa
## 5 setosa
## 6 setosa
## 7 setosa
## 8 setosa
## 9 setosa
## 10 setosa
## 11 setosa
## 12 setosa
## 13 setosa
## 14 setosa
## 15 setosa
## 16 setosa
## 17 setosa
## 18 setosa
## 19 setosa
## 20 setosa
## 21 setosa
## 22 setosa
## 23 setosa
## 24 setosa
## 25 setosa
## 26 setosa
## 27 setosa
## 28 setosa
## 29 setosa
## 30 setosa
## 31 setosa
## 32 setosa
## 33 setosa
## 34 setosa
## 35 setosa
## 36 setosa
## 37 setosa
## 38 setosa
## 39 setosa
## 40 setosa
## 41 setosa
## 42 setosa
## 43 setosa
## 44 setosa
## 45 setosa
## 46 setosa
## 47 setosa
## 48 setosa
## 49 setosa
## 50 setosa
## 51 versicolor
## 52 versicolor
## 53 versicolor
## 54 versicolor
## 55 versicolor
## 56 versicolor
## 57 versicolor
## 58 versicolor
## 59 versicolor
## 60 versicolor
## 61 versicolor
## 62 versicolor
## 63 versicolor
## 64 versicolor
## 65 versicolor
## 66 versicolor
## 67 versicolor
## 68 versicolor
## 69 versicolor
## 70 versicolor
## 71 versicolor
## 72 versicolor
## 73 versicolor
## 74 versicolor
## 75 versicolor
## 76 versicolor
## 77 versicolor
## 78 versicolor
## 79 versicolor
## 80 versicolor
## 81 versicolor
## 82 versicolor
## 83 versicolor
## 84 versicolor
## 85 versicolor
## 86 versicolor
## 87 versicolor
## 88 versicolor
## 89 versicolor
## 90 versicolor
## 91 versicolor
## 92 versicolor
## 93 versicolor
## 94 versicolor
## 95 versicolor
## 96 versicolor
## 97 versicolor
## 98 versicolor
## 99 versicolor
## 100 versicolor
## 101 virginica
## 102 virginica
## 103 virginica
## 104 virginica
## 105 virginica
## 106 virginica
## 107 virginica
## 108 virginica
## 109 virginica
## 110 virginica
## 111 virginica
## 112 virginica
## 113 virginica
## 114 virginica
## 115 virginica
## 116 virginica
## 117 virginica
## 118 virginica
## 119 virginica
## 120 virginica
## 121 virginica
## 122 virginica
## 123 virginica
## 124 virginica
## 125 virginica
## 126 virginica
## 127 virginica
## 128 virginica
## 129 virginica
## 130 virginica
## 131 virginica
## 132 virginica
## 133 virginica
## 134 virginica
## 135 virginica
## 136 virginica
## 137 virginica
## 138 virginica
## 139 virginica
## 140 virginica
## 141 virginica
## 142 virginica
## 143 virginica
## 144 virginica
## 145 virginica
## 146 virginica
## 147 virginica
## 148 virginica
## 149 virginica
## 150 virginica
## Sepal.Length Sepal.Width Petal.Length Petal.Width
## 1 5.1 3.5 1.4 0.2
## 2 4.9 3.0 1.4 0.2
## 3 4.7 3.2 1.3 0.2
## 4 4.6 3.1 1.5 0.2
## 5 5.0 3.6 1.4 0.2
## 6 5.4 3.9 1.7 0.4
## 7 4.6 3.4 1.4 0.3
## 8 5.0 3.4 1.5 0.2
## 9 4.4 2.9 1.4 0.2
## 10 4.9 3.1 1.5 0.1
## 11 5.4 3.7 1.5 0.2
## 12 4.8 3.4 1.6 0.2
## 13 4.8 3.0 1.4 0.1
## 14 4.3 3.0 1.1 0.1
## 15 5.8 4.0 1.2 0.2
## 16 5.7 4.4 1.5 0.4
## 17 5.4 3.9 1.3 0.4
## 18 5.1 3.5 1.4 0.3
## 19 5.7 3.8 1.7 0.3
## 20 5.1 3.8 1.5 0.3
## 21 5.4 3.4 1.7 0.2
## 22 5.1 3.7 1.5 0.4
## 23 4.6 3.6 1.0 0.2
## 24 5.1 3.3 1.7 0.5
## 25 4.8 3.4 1.9 0.2
## 26 5.0 3.0 1.6 0.2
## 27 5.0 3.4 1.6 0.4
## 28 5.2 3.5 1.5 0.2
## 29 5.2 3.4 1.4 0.2
## 30 4.7 3.2 1.6 0.2
## 31 4.8 3.1 1.6 0.2
## 32 5.4 3.4 1.5 0.4
## 33 5.2 4.1 1.5 0.1
## 34 5.5 4.2 1.4 0.2
## 35 4.9 3.1 1.5 0.2
## 36 5.0 3.2 1.2 0.2
## 37 5.5 3.5 1.3 0.2
## 38 4.9 3.6 1.4 0.1
## 39 4.4 3.0 1.3 0.2
## 40 5.1 3.4 1.5 0.2
## 41 5.0 3.5 1.3 0.3
## 42 4.5 2.3 1.3 0.3
## 43 4.4 3.2 1.3 0.2
## 44 5.0 3.5 1.6 0.6
## 45 5.1 3.8 1.9 0.4
## 46 4.8 3.0 1.4 0.3
## 47 5.1 3.8 1.6 0.2
## 48 4.6 3.2 1.4 0.2
## 49 5.3 3.7 1.5 0.2
## 50 5.0 3.3 1.4 0.2
## 51 7.0 3.2 4.7 1.4
## 52 6.4 3.2 4.5 1.5
## 53 6.9 3.1 4.9 1.5
## 54 5.5 2.3 4.0 1.3
## 55 6.5 2.8 4.6 1.5
## 56 5.7 2.8 4.5 1.3
## 57 6.3 3.3 4.7 1.6
## 58 4.9 2.4 3.3 1.0
## 59 6.6 2.9 4.6 1.3
## 60 5.2 2.7 3.9 1.4
## 61 5.0 2.0 3.5 1.0
## 62 5.9 3.0 4.2 1.5
## 63 6.0 2.2 4.0 1.0
## 64 6.1 2.9 4.7 1.4
## 65 5.6 2.9 3.6 1.3
## 66 6.7 3.1 4.4 1.4
## 67 5.6 3.0 4.5 1.5
## 68 5.8 2.7 4.1 1.0
## 69 6.2 2.2 4.5 1.5
## 70 5.6 2.5 3.9 1.1
## 71 5.9 3.2 4.8 1.8
## 72 6.1 2.8 4.0 1.3
## 73 6.3 2.5 4.9 1.5
## 74 6.1 2.8 4.7 1.2
## 75 6.4 2.9 4.3 1.3
## 76 6.6 3.0 4.4 1.4
## 77 6.8 2.8 4.8 1.4
## 78 6.7 3.0 5.0 1.7
## 79 6.0 2.9 4.5 1.5
## 80 5.7 2.6 3.5 1.0
## 81 5.5 2.4 3.8 1.1
## 82 5.5 2.4 3.7 1.0
## 83 5.8 2.7 3.9 1.2
## 84 6.0 2.7 5.1 1.6
## 85 5.4 3.0 4.5 1.5
## 86 6.0 3.4 4.5 1.6
## 87 6.7 3.1 4.7 1.5
## 88 6.3 2.3 4.4 1.3
## 89 5.6 3.0 4.1 1.3
## 90 5.5 2.5 4.0 1.3
## 91 5.5 2.6 4.4 1.2
## 92 6.1 3.0 4.6 1.4
## 93 5.8 2.6 4.0 1.2
## 94 5.0 2.3 3.3 1.0
## 95 5.6 2.7 4.2 1.3
## 96 5.7 3.0 4.2 1.2
## 97 5.7 2.9 4.2 1.3
## 98 6.2 2.9 4.3 1.3
## 99 5.1 2.5 3.0 1.1
## 100 5.7 2.8 4.1 1.3
## 101 6.3 3.3 6.0 2.5
## 102 5.8 2.7 5.1 1.9
## 103 7.1 3.0 5.9 2.1
## 104 6.3 2.9 5.6 1.8
## 105 6.5 3.0 5.8 2.2
## 106 7.6 3.0 6.6 2.1
## 107 4.9 2.5 4.5 1.7
## 108 7.3 2.9 6.3 1.8
## 109 6.7 2.5 5.8 1.8
## 110 7.2 3.6 6.1 2.5
## 111 6.5 3.2 5.1 2.0
## 112 6.4 2.7 5.3 1.9
## 113 6.8 3.0 5.5 2.1
## 114 5.7 2.5 5.0 2.0
## 115 5.8 2.8 5.1 2.4
## 116 6.4 3.2 5.3 2.3
## 117 6.5 3.0 5.5 1.8
## 118 7.7 3.8 6.7 2.2
## 119 7.7 2.6 6.9 2.3
## 120 6.0 2.2 5.0 1.5
## 121 6.9 3.2 5.7 2.3
## 122 5.6 2.8 4.9 2.0
## 123 7.7 2.8 6.7 2.0
## 124 6.3 2.7 4.9 1.8
## 125 6.7 3.3 5.7 2.1
## 126 7.2 3.2 6.0 1.8
## 127 6.2 2.8 4.8 1.8
## 128 6.1 3.0 4.9 1.8
## 129 6.4 2.8 5.6 2.1
## 130 7.2 3.0 5.8 1.6
## 131 7.4 2.8 6.1 1.9
## 132 7.9 3.8 6.4 2.0
## 133 6.4 2.8 5.6 2.2
## 134 6.3 2.8 5.1 1.5
## 135 6.1 2.6 5.6 1.4
## 136 7.7 3.0 6.1 2.3
## 137 6.3 3.4 5.6 2.4
## 138 6.4 3.1 5.5 1.8
## 139 6.0 3.0 4.8 1.8
## 140 6.9 3.1 5.4 2.1
## 141 6.7 3.1 5.6 2.4
## 142 6.9 3.1 5.1 2.3
## 143 5.8 2.7 5.1 1.9
## 144 6.8 3.2 5.9 2.3
## 145 6.7 3.3 5.7 2.5
## 146 6.7 3.0 5.2 2.3
## 147 6.3 2.5 5.0 1.9
## 148 6.5 3.0 5.2 2.0
## 149 6.2 3.4 5.4 2.3
## 150 5.9 3.0 5.1 1.8
## Species
## 1 setosa
## 2 setosa
## 3 setosa
## 4 setosa
## 5 setosa
## 6 setosa
## 7 setosa
## 8 setosa
## 9 setosa
## 10 setosa
## 11 setosa
## 12 setosa
## 13 setosa
## 14 setosa
## 15 setosa
## 16 setosa
## 17 setosa
## 18 setosa
## 19 setosa
## 20 setosa
## 21 setosa
## 22 setosa
## 23 setosa
## 24 setosa
## 25 setosa
## 26 setosa
## 27 setosa
## 28 setosa
## 29 setosa
## 30 setosa
## 31 setosa
## 32 setosa
## 33 setosa
## 34 setosa
## 35 setosa
## 36 setosa
## 37 setosa
## 38 setosa
## 39 setosa
## 40 setosa
## 41 setosa
## 42 setosa
## 43 setosa
## 44 setosa
## 45 setosa
## 46 setosa
## 47 setosa
## 48 setosa
## 49 setosa
## 50 setosa
## 51 versicolor
## 52 versicolor
## 53 versicolor
## 54 versicolor
## 55 versicolor
## 56 versicolor
## 57 versicolor
## 58 versicolor
## 59 versicolor
## 60 versicolor
## 61 versicolor
## 62 versicolor
## 63 versicolor
## 64 versicolor
## 65 versicolor
## 66 versicolor
## 67 versicolor
## 68 versicolor
## 69 versicolor
## 70 versicolor
## 71 versicolor
## 72 versicolor
## 73 versicolor
## 74 versicolor
## 75 versicolor
## 76 versicolor
## 77 versicolor
## 78 versicolor
## 79 versicolor
## 80 versicolor
## 81 versicolor
## 82 versicolor
## 83 versicolor
## 84 versicolor
## 85 versicolor
## 86 versicolor
## 87 versicolor
## 88 versicolor
## 89 versicolor
## 90 versicolor
## 91 versicolor
## 92 versicolor
## 93 versicolor
## 94 versicolor
## 95 versicolor
## 96 versicolor
## 97 versicolor
## 98 versicolor
## 99 versicolor
## 100 versicolor
## 101 virginica
## 102 virginica
## 103 virginica
## 104 virginica
## 105 virginica
## 106 virginica
## 107 virginica
## 108 virginica
## 109 virginica
## 110 virginica
## 111 virginica
## 112 virginica
## 113 virginica
## 114 virginica
## 115 virginica
## 116 virginica
## 117 virginica
## 118 virginica
## 119 virginica
## 120 virginica
## 121 virginica
## 122 virginica
## 123 virginica
## 124 virginica
## 125 virginica
## 126 virginica
## 127 virginica
## 128 virginica
## 129 virginica
## 130 virginica
## 131 virginica
## 132 virginica
## 133 virginica
## 134 virginica
## 135 virginica
## 136 virginica
## 137 virginica
## 138 virginica
## 139 virginica
## 140 virginica
## 141 virginica
## 142 virginica
## 143 virginica
## 144 virginica
## 145 virginica
## 146 virginica
## 147 virginica
## 148 virginica
## 149 virginica
## 150 virginica
Slide with Bullets
- Bullet 1
- Bullet 2
- Bullet 3
- Bullet 1
- Bullet 2
- Bullet 3
Slide with R Output
## speed dist
## Min. : 4.0 Min. : 2.00
## 1st Qu.:12.0 1st Qu.: 26.00
## Median :15.0 Median : 36.00
## Mean :15.4 Mean : 42.98
## 3rd Qu.:19.0 3rd Qu.: 56.00
## Max. :25.0 Max. :120.00
Slide with Plot
